Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

10 paolo mancosu4) What is the role of analogy and other types of inductive reasoning inmathematics? Can Bayesianism be applied to mathematics?5) What is the relationship between diagrammatic thinking and formalreasoning? How to account for the fruitfulness of diagrammatic reasoningin algebraic topology?Of course, several of these issues had already been discussed in the literaturebefore Corfield, but his book was the first to bring them together. Thus,Corfield’s proposed philosophy of mathematics displays the three featuresof the mavericks’ approach mentioned at the outset. In comparison withprevious contributions in that tradition, he expands the set of topics that canbe fruitfully investigated and seems to be less concerned than Lakatos andKitcher with providing a grand theory of mathematical change. His emphasisis on more localized case studies. The foundationalist and the analytic traditionin philosophy of mathematics are dismissed as irrelevant in addressing themost pressing problems for a ‘real’ philosophy of mathematics. In Section 5, Iwill comment on how Corfield’s program relates to the contributions in thisvolume.3 Maddy on mathematical practiceFaithfulness to mathematical practice is for Maddy a criterion of adequacy for asatisfactory philosophy of mathematics (Maddy, 1990, p.23 and p. 28). In her1990 book, Realism in Mathematics, she took her start from Quine’s naturalizedepistemology (there is no first philosophy, natural science is the court ofarbitration even for its own methodology) and forms of the indispensabilityargument. Her realism originated from a combination of Quine’s Platonismwith that of Gödel. But Maddy is also critical of certain aspects of Quine’s andGödel’s Platonisms, for she claims that both fail to capture certain aspects of themathematical experience. In particular, she finds objectionable that unappliedmathematics is not granted right of citizenship in Quine’s account (see Quine,1984, p.788) and,contra Quine, she emphasizes the autonomy of mathematicsfrom physics. By contrast, the Gödelian brand of Platonism respects theautonomy of mathematics but its weakness consists in the postulation of afaculty of intuition in analogy with perception in the natural sciences. Gödelappealed to such a faculty of intuition to account for those parts of mathematicswhich can be given an ‘intrinsic’ justification. However, there are parts ofmathematics for which such ‘intrinsic’, intuitive, justifications cannot be givenand for those one appeals to ‘extrinsic’ justifications; that is, a justification in